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YEAR 11
REVISION BLASTER
UNIT 2 STAGE 2
GCSE MATHS REVISION
UNIT 2 (stage 2) – all about:SHAPE & ALGEBRA & NUMBER• Circle theorems• Angles & lines & shapes• Simplifying & factorising• Linear graphs (y=mx±c)• Percentages
CIRCLE THEOREMS
The angle made at the centre of a circle (by an arc or chord) is double the angle made at the circumference by the same arc or chord.
2xo
2xo 2xo 2xo
2xo 2xo 2xo 2xo
xo
xo
xoxo
xo xo xo xo
o oo o
o o o o
Angle at centre is a reflex angle
Another Theorem
The Angle at Centre (Arrow) TheoremTheorem 1
o Diameter
90o angle in a semi-circle90o angle in a semi-circle20o angle sum triangle
90o angle in a semi-circle
e
o
a
b
c
70o
d
30o
Find the unknown angles below stating a reason.
angle a = angle b = angle c = angle d = angle e =
60o angle sum triangle
Angle in Semi-Circle TheoremTheorem 2
This is a special case of Theorem 1: angle in a semi-circle is always 90°
Th2
Angles subtended by an arc or chord in the same segment are equal.
xo xo
xo
xo
xo
yo
yo
Th3
The Chord (Bow) TheoremTheorem 3
The angle between a tangent and a radius is 90o. (Tan/rad)
o
The Tangent TheoremTheorem 4
The Alternate Segment TheoremTheorem 5
The angle between a tangent and a chord through the point of contact is equal to the angle made by that chord in the alternate segment.
xo
xo
yo
yo
45o (Alt Seg)
60o (Alt Seg)
75o angle sum triangle
45o
xo
yo
60o
zo
Find the missing angles below giving reasons in each case.
angle x = angle y = angle z =
Th5
Cyclic Quadrilateral TheoremTheorem 6
The opposite angles of a cyclic quadrilateral are supplementary. (They sum to 180o)
w
y
x
z
Angles x + z = 180o
Angles y + w = 180o
r
p
s
q
Angles p + r = 180o
Angles q + s = 180o
All corners touch the circle circumference
The Two Tangent TheoremTheorem 7
From any point outside a circle only two tangents can be drawn and they are equal in length.
P
T
UQ
R
PT = PQ
P
T
U
Q
R
PT = PQ
Th7
Circle Theorem Answers
1. (a) 58°(b) 2 angles are the same (=58°)
2. (a) (i) 70°(opposite angles of) cyclic quadrilateral
(ii) 140°angle at centre is twice angle at circumference
3. (a) (i) 40°(ii) 140°
(b) Right angle at A / C (Tangent Theorem): at B 2 x 50° angles
4. x = 52°; y = 76° and z = 58°
Angles
Which angles are equal to each other?
ab
cd
ef
gh
Alternate angles are
equal
Alternate angles are
equal
a
b
a = ba = b
Look for an F-shape
Look for a Z-shape
Corresponding angles are
equal
Corresponding angles are
equal
a
b
a = ba = b
Look for a C- or U-shape
Interior angles add up to 180°
Interior angles add up to 180°
a
b
a + b = 180°a + b = 180°
ab
c
Any exterior angle in a triangle is equal to the sum of the two opposite interior
angles.
Any exterior angle in a triangle is equal to the sum of the two opposite interior
angles.
a = b + c
We can prove this by constructing a line parallel to this side.These alternate angles are equal.These corresponding angles are equal.
bc
Calculate the size of angle a.
a28º
45º
Hint: Add another parallel
line.
Hint: Add another parallel
line.
a = 28º + 45º = 73º
Angles & Lines & Shapes Answers
1. (a) x = 60.5(b) Not parallel because angle A is not equal to x,
or because angle B is not equal to x,or because alternate (or Z) angles are not equal,or because corresponding angles are not equal.
2. (a) (i) x = 74
(ii) y = 36°(b) No. Because triangle ABD is not isosceles.
or Because y is not equal to 38°.3. (a) 30°
(b) 40°
4. a = 50° and b = 110°
AD BREAKYear 11 Unit 2 Exam(s). Monday 15th November-
NEXT WEEK!!
Don’t Forget: CALCULATORS
PEN
PENCIL
RULER
REVISE
Algebra
Expanding and simplifying
• (x+2)(x+3)
x² +3x
+2x +6
= x²
= x² + 5x + 6
+ 3x+ 2x +6
Factorising
• x² + 7x + 12
+12 x
x
+6
+2
+6x
+2x
= x² + 8x +12
= x² + 6x +2x +12
+4
+3
+4x
+3x
= x² + 4x +3x +12
= x² + 7x +12
So,
x² + 7x + 12 = ( )( )
Which simplifies to: 20a³b³
• 4ab x 5a²b²Simplifying
10ab²
First, focus on the 4a³b x 5a²b²
Rearranging gives: 4x5 x axa² x bxb²
20a³b³
10ab²
= 20 a³ b³
10 a b²= 2a²b
Simplifying & Factorising Answers
1. (a) 3x(x – 2y)
(b) (y – 7)(y – 2)
2. (a) (x – 7)(x – 3)
3. (a) 24abc
4. a4 c–1 or
5. (b) 5xy(x + 3y2)
6. (b) 2a2 b2 c
c
a4
STRAIGHT LINE GRAPHS
General equation of straight line.
y = mx + c
y = mx + c
Gradient/ Steepness of the line.
Y-intercept. (where the graph crosses the y axis)
Plotting a straight line…y = 2x -1Pick 3 values of x. (e.g -2, 0, 2)
SUBSTITUTE these into the equation (y=2x-1)
xy
when x = -2,
Y = (2x-2) – 1
Y = - 4 – 1
Y = -5
When x = 0,
Y = (2x0) – 1
Y = 0 – 1
Y = -1
When x = 2
Y = (2x2) – 1
Y = 4 – 1
Y = 3
-2 0 2
-5 -1 3
Find the gradient:
Draw a triangle under the graph.
Make sure the height and width of the triangle are WHOLE numbers
Rise = 8
Run = 4
Gradient = Rise = 8
Run = 4
= 2
Plot these pairs of co-ordinates. Join up the points, and extend across the axes.
y =
Gradient/ Steepness of the line.
Y-intercept. (where the graph crosses the y axis)
mx+ c
2 -1
Key Facts•Parallel lines have the SAME
gradient
• Graphs with a negative gradient slope DOWNHILL
•Graphs with a positive gradient slope UPHILL
•The bigger the gradient the steeper the slope
Linear Graphs Answers
1. 0.8, 4/5, or equivalent2. (a) y = 0.4x + 18
(b) 583. (a) (– 6, 0)
(b) ½(c) Line drawn parallel to 2y = x + 6
4. (i) (–½, 0)(ii) reasonably parallel line crossing y axis below origin(iii) parallel lines do not meet
Percentage Increase
and Decrease
Increase £20 by 15%
Method 1Find 15%10% = £2 5% = £1 so 15% = £3
Now add it on to £20£20 + £3 = £23
Increase by 15% so
1.15 is the multiplier
Method 2Method 2
I need to find 115% I need to find 115% OF original amountOF original amount115% of £20 is 115% of £20 is 1.15 x £20 = £231.15 x £20 = £23
Decrease £60 by 5%
Method 1
Find 5% of £60
10% = £6 5% = £3
Subtract from £60
£60 - £3 = £57
Method 2
A decrease of 5% is same as 95% of original amount
So 0.95 x £60 = £57 95% of original amount means 0.95 is the multiplier
Compound Interest
• £1,000 in bank earns 5% interest per year. How much will you have in 3 years?
After 1 year 1000 x 1.05 = 1050
After 2 years 1050 x 1.05= 1102.50
After 3 years 1102.50 x 1.05 = 1157.63
OR 1000 x 1.05 x 1.05 x1.05=
1000 x 1.05³=£1157.63
1.05 is the multiplier!
Percentages Answers
1.£2140 (allow £140)
2.(a) 1.029
(b) 1 087 401 937 (allow 1 087 000 000)
3.1st year is £20; 2nd year is £20.80
Interest is £40.80 so Amir is wrong
4.(a) 2.04
(b) 6 (windmills)